The Anti-Unitarity of Time Reversal & Co-representations of Lorentzian Pin Groups

Abstract

In the representation theory of Lorentzian orthogonal groups, there are well known arguments as to why the parity inversion operator $\mathcal{P}$ and the time reversal operator $\mathcal{T}$, should be realized as linear and anti-linear operators respectively (Wigner 1932). Despite this, standard constructions of double covers of the Lorentzian orthogonal groups naturally build time reversal operators in such a manner that they are linear, and the anti-linearity is put in ad-hoc after the fact. This article introduces a viewpoint naturally incorporating the anti-linearity into the construction of these double covers, through what Wigner called co-representations, a kind of semi-linear representation. It is shown how the standard spinoral double covers of the Lorentz group -- $\operatorname{Pin}(1,3)$ and $\operatorname{Pin}(3,1)$ -- may be naturally centrally extended for this purpose, and the relationship between the $\mathcal{C}$, $\mathcal{P}$, and $\mathcal{T}$ operators is discussed. Additionally a mapping is constructed demonstrating an interesting equivalence between Majorana and Weyl spinors. Finally a co-representation is built for the de Sitter group $\operatorname{Pin}(1,4)$, and it is demonstrated how a theory with this symmetry has no truly scalar fermion mass terms.